Neighbor Locating Coloring on Graphs: Three Products

Let G be a simple finite graph. A k -coloring of G is a partition π = { S 1 , ⋯ , S k } of V ( G ) so that each S i is an independent set and any vertex in S i takes color i . A k -coloring π = { S 1 , ⋯ , S k } of V ( G ) is a neighbor locating coloring if for any two vertices u , v ∈ S i , there is a color class S j for which one of them has a neighbor in S j and the other does not. The minimum k with this property is said to be the neighbor locating chromatic number of G , denoted by χ N L ( G ) . We initiate the study of the neighbor locating coloring of graphs resulting from three types of product of two graphs. We investigate the neighbor locating chromatic number of Cartesian, lexicographic, and corona products of two graphs. Finally, we untangle the neighbor locating chromatic number of any of the aforementioned three products of cycles, paths, and complete graphs.